One-to-one and Inverse
Functions
Review:
A
is any set of ordered pairs.
A
is a set of ordered pairs where
x is not repeated.
A
repeated.
function does not have any y values
Only
functions.
functions can have
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2
What is an Inverse?
An inverse relation is a relation that performs the
opposite operation on x (the domain).
Examples:
f(x) = x – 3
g(x) = x , x ≥ 0
h(x) = 2x
k(x) = -x + 3
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f-1(x) = x + 3
g-1(x) = x2 , x ≥ 0
h-1(x) = ½ x
k-1(x)= -(x – 3)
3
Section 1.9 : Illustration
of the Definition of Inverse
Functions
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The ordered pairs of the function f are reversed to
produce the ordered pairs of the inverse relation.
Example: Given the function
f = {(1, 1), (2, 3), (3, 1), (4, 2)}, its domain is {1, 2, 3, 4}
and its range is {1, 2, 3}.
The inverse
of f is {(1, 1), (3, 2), (1, 3), (2, 4)}.
The domain of the inverse relation is the range of the
original function.
The range of the inverse relation is the domain of the
original function.
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How do we know if an inverse
function exists?
• Inverse functions only exist if the original function
is one to one. Otherwise it is an inverse relation
and cannot be written as f-1(x).
• What does it mean to be one to one?
That there are no repeated y values.
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6
Horizontal Line Test
Used to test if a function is one-to one
If the line intersection more than once then it is not one to
one.
Therefore there is not inverse function.
Example: The function
y = x2 – 4x + 7 is not one-to-one
because a horizontal line can
intersect the graph twice.
Examples points: (0, 7) & (4, 7).
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y
(4, 7)
(0, 7)
y=7
2
x
2
7
Example: Apply the horizontal line test to the graphs
below to determine if the functions are one-to-one.
b) y = x3 + 3x2 – x – 1
a) y = x3
y
-4
y
8
8
4
4
4
-4
x
one-to-one
The Inverse is a Function
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4
x
not one-to-one
The Inverse is a Relation
8
The graphs of a relation and its inverse are reflections
in the line y = x.
Example: Find the graph of the inverse relation
geometrically from the graph of f (x) = 1 ( x 3  2)
4
y
The ordered pairs of f are given by
the equation y  1 ( x 3  2) .
2
4
The ordered pairs of the inverse are
given by x  1 ( y 3  2) .
x
-2
4
( y  2)
x
4
3
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y=x
2
-2
( x 3  2)
y
4
9
Section 1.9 : Figure 1.93, Graph
of an Inverse Function
Functions and their
inverses are symmetric
over the line y =x
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To find the inverse of a relation algebraically,
interchange x and y and solve for y.
Example: Find the inverse relation algebraically for the
function f (x) = 3x + 2.
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DETERMINING IF 2 FUNCTIONS ARE INVERSES:
The inverse function “undoes” the original function,
that is, f -1( f (x)) = x.
The function is the inverse of its inverse function,
that is, f ( f -1(x)) = x.
Example: The inverse of f (x) = x3 is f -1(x) = 3 x .
f
-1(
3
3
f(x)) = x = x and f ( f -1(x)) = ( 3 x )3 = x.
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x 1
Example: Verify that the function g(x) =
is the inverse of f(x) = 2x – 1. 2
( f ( x)  1) (( 2 x  1)  1)
2x
g( f(x)) =
=
=
=x
2
2
2
f(g(x)) = 2g(x) – 1 = 2( x  1 ) – 1 = (x + 1) – 1 = x
2
It follows that g = f -1.
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Now Try: Page 99 #13, 15, 23
pg 101 # 69, 71, 73
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Review of Today’s Material
• A function must be 1-1 (pass the horizontal
line test) to have an inverse function (written
f-1(x)) otherwise the inverse is a relation (y =)
• To find an inverse: 1) Switch x and y
2) Solve for y
• Original and Inverses are symmetric over y =x
• “
“
” have reverse domain & ranges
• Given two relations to test for inverses.
f(f-1(x)) = x and f-1(f(x)) = x **both must be true**
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Practice Problems and Homework
• Page 99-100
# 16, 18, 20, 24, 39, 41, 43
#55-67 odd
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